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Some Controllable and Uncontrollable Degenerate Four-Level Quantum Systems

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Abstract

We analyze the controllability of three quantum systems that belong to a specific class of four-level quantum systems with twice degenerate highest excited energy level and with forbidden transition between the two remaining non-degenerate levels. For this purpose we perform numerical computation, construct the dynamical Lie algebras generated by all commutators of the free and interaction Hamiltonians, and show that two quantum systems are irreducible and controllable while the third system is reducible and hence uncontrollable. The reducibility and uncontrollability are proved by constructing a conserved Hermitian operator (physical quantity). The controllability is proved by constructing the dynamical Lie algebra and showing that it has maximal rank. These findings indicate that depending on the values of certain particular matrix entries of the interaction Hamiltonian, quantum systems in the class under consideration can be either uncontrollable or controllable.

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Funding

This work was supported by the Russian Science Foundation under grant no. 22-11-00330, https://rscf.ru/project/22-11-00330/.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Anastasia A. Myachkova & Alexander N. Pechen

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  1. Anastasia A. Myachkova
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  2. Alexander N. Pechen
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Correspondence to Alexander N. Pechen.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 321, pp. 237–251 https://doi.org/10.4213/tm4321.

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Myachkova, A.A., Pechen, A.N. Some Controllable and Uncontrollable Degenerate Four-Level Quantum Systems. Proc. Steklov Inst. Math. 321, 222–235 (2023). https://doi.org/10.1134/S0081543823020165

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